LMFDB - Embedded newform 4232.2.a.u.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4232,2,Mod(1,4232)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4232.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4232, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4232 = 2^{3} \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4232.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,8,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$33.7926901354$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{24})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 1 $ x^4 - 4*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.517638$ of defining polynomial
Character	$\chi$	$=$	4232.1

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421 q^{3} +2.51764 q^{5} -0.732051 q^{7} -1.00000 q^{9} +0.732051 q^{11} -4.86370 q^{13} +3.56048 q^{15} -6.44949 q^{17} -5.56048 q^{19} -1.03528 q^{21} +1.33850 q^{25} -5.65685 q^{27} +6.42418 q^{29} +2.49938 q^{31} +1.03528 q^{33} -1.84304 q^{35} +2.74202 q^{37} -6.87832 q^{39} +2.46410 q^{41} +2.02922 q^{43} -2.51764 q^{45} -8.71279 q^{47} -6.46410 q^{49} -9.12096 q^{51} +3.08933 q^{53} +1.84304 q^{55} -7.86370 q^{57} -11.4999 q^{59} -2.56753 q^{61} +0.732051 q^{63} -12.2450 q^{65} -13.5347 q^{67} -13.2414 q^{71} -7.26670 q^{73} +1.89293 q^{75} -0.535898 q^{77} +17.1210 q^{79} -5.00000 q^{81} +5.50543 q^{83} -16.2375 q^{85} +9.08516 q^{87} -5.69578 q^{89} +3.56048 q^{91} +3.53465 q^{93} -13.9993 q^{95} +14.9306 q^{97} -0.732051 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{5} + 4 q^{7} - 4 q^{9} - 4 q^{11} - 4 q^{13} - 4 q^{15} - 16 q^{17} - 4 q^{19} + 4 q^{25} - 8 q^{29} - 8 q^{31} + 8 q^{35} - 24 q^{37} - 8 q^{39} - 4 q^{41} - 8 q^{45} - 8 q^{47} - 12 q^{49}+ \cdots + 4 q^{99}+O(q^{100}) $ 4 * q + 8 * q^5 + 4 * q^7 - 4 * q^9 - 4 * q^11 - 4 * q^13 - 4 * q^15 - 16 * q^17 - 4 * q^19 + 4 * q^25 - 8 * q^29 - 8 * q^31 + 8 * q^35 - 24 * q^37 - 8 * q^39 - 4 * q^41 - 8 * q^45 - 8 * q^47 - 12 * q^49 - 8 * q^55 - 16 * q^57 - 8 * q^59 - 4 * q^63 - 16 * q^65 - 32 * q^67 - 16 * q^75 - 16 * q^77 + 32 * q^79 - 20 * q^81 + 8 * q^83 - 44 * q^85 + 24 * q^87 - 16 * q^89 - 4 * q^91 - 8 * q^93 + 16 * q^97 + 4 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$. You can download additional coefficients here.

Currently showing only $a_p$; display all $a_n$ Currently showing all $a_n$; display only $a_p$

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.41421	0.816497	0.408248	−	0.912871i	$-0.366140\pi$
$3$	1.41421	0.816497	0.408248	+	0.912871i	$0.366140\pi$
$4$	0	0
$5$	2.51764	1.12592	0.562961	−	0.826483i	$-0.309662\pi$
$5$	2.51764	1.12592	0.562961	+	0.826483i	$0.309662\pi$
$6$	0	0
$7$	−0.732051	−0.276689	−0.138345	−	0.990384i	$-0.544178\pi$
$7$	−0.732051	−0.276689	−0.138345	+	0.990384i	$0.544178\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	0.732051	0.220722	0.110361	−	0.993892i	$-0.464799\pi$
$11$	0.732051	0.220722	0.110361	+	0.993892i	$0.464799\pi$
$12$	0	0
$13$	−4.86370	−1.34895	−0.674474	−	0.738298i	$-0.735630\pi$
$13$	−4.86370	−1.34895	−0.674474	+	0.738298i	$0.735630\pi$
$14$	0	0
$15$	3.56048	0.919311
$16$	0	0
$17$	−6.44949	−1.56423	−0.782116	−	0.623133i	$-0.785859\pi$
$17$	−6.44949	−1.56423	−0.782116	+	0.623133i	$0.785859\pi$
$18$	0	0
$19$	−5.56048	−1.27566	−0.637831	−	0.770177i	$-0.720168\pi$
$19$	−5.56048	−1.27566	−0.637831	+	0.770177i	$0.720168\pi$
$20$	0	0
$21$	−1.03528	−0.225916
$22$	0	0
$23$	0	0
$23$	0	0
$24$	0	0
$25$	1.33850	0.267700
$26$	0	0
$27$	−5.65685	−1.08866
$28$	0	0
$29$	6.42418	1.19294	0.596470	−	0.802635i	$-0.296569\pi$
$29$	6.42418	1.19294	0.596470	+	0.802635i	$0.296569\pi$
$30$	0	0
$31$	2.49938	0.448902	0.224451	−	0.974485i	$-0.427941\pi$
$31$	2.49938	0.448902	0.224451	+	0.974485i	$0.427941\pi$
$32$	0	0
$33$	1.03528	0.180218
$34$	0	0
$35$	−1.84304	−0.311530
$36$	0	0
$37$	2.74202	0.450785	0.225393	−	0.974268i	$-0.427634\pi$
$37$	2.74202	0.450785	0.225393	+	0.974268i	$0.427634\pi$
$38$	0	0
$39$	−6.87832	−1.10141
$40$	0	0
$41$	2.46410	0.384828	0.192414	−	0.981314i	$-0.438368\pi$
$41$	2.46410	0.384828	0.192414	+	0.981314i	$0.438368\pi$
$42$	0	0
$43$	2.02922	0.309454	0.154727	−	0.987957i	$-0.450550\pi$
$43$	2.02922	0.309454	0.154727	+	0.987957i	$0.450550\pi$
$44$	0	0
$45$	−2.51764	−0.375307
$46$	0	0
$47$	−8.71279	−1.27089	−0.635446	−	0.772145i	$-0.719184\pi$
$47$	−8.71279	−1.27089	−0.635446	+	0.772145i	$0.719184\pi$
$48$	0	0
$49$	−6.46410	−0.923443
$50$	0	0
$51$	−9.12096	−1.27719
$52$	0	0
$53$	3.08933	0.424352	0.212176	−	0.977231i	$-0.431945\pi$
$53$	3.08933	0.424352	0.212176	+	0.977231i	$0.431945\pi$
$54$	0	0
$55$	1.84304	0.248515
$56$	0	0
$57$	−7.86370	−1.04157
$58$	0	0
$59$	−11.4999	−1.49716	−0.748579	−	0.663045i	$-0.769264\pi$
$59$	−11.4999	−1.49716	−0.748579	+	0.663045i	$0.769264\pi$
$60$	0	0
$61$	−2.56753	−0.328738	−0.164369	−	0.986399i	$-0.552559\pi$
$61$	−2.56753	−0.328738	−0.164369	+	0.986399i	$0.552559\pi$
$62$	0	0
$63$	0.732051	0.0922297
$64$	0	0
$65$	−12.2450	−1.51881
$66$	0	0
$67$	−13.5347	−1.65352	−0.826761	−	0.562554i	$-0.809819\pi$
$67$	−13.5347	−1.65352	−0.826761	+	0.562554i	$0.809819\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−13.2414	−1.57146	−0.785732	−	0.618567i	$-0.787714\pi$
$71$	−13.2414	−1.57146	−0.785732	+	0.618567i	$0.787714\pi$
$72$	0	0
$73$	−7.26670	−0.850503	−0.425252	−	0.905075i	$-0.639814\pi$
$73$	−7.26670	−0.850503	−0.425252	+	0.905075i	$0.639814\pi$
$74$	0	0
$75$	1.89293	0.218576
$76$	0	0
$77$	−0.535898	−0.0610713
$78$	0	0
$79$	17.1210	1.92626	0.963129	−	0.269040i	$-0.0867064\pi$
$79$	17.1210	1.92626	0.963129	+	0.269040i	$0.0867064\pi$
$80$	0	0
$81$	−5.00000	−0.555556
$82$	0	0
$83$	5.50543	0.604299	0.302150	−	0.953260i	$-0.402296\pi$
$83$	5.50543	0.604299	0.302150	+	0.953260i	$0.402296\pi$
$84$	0	0
$85$	−16.2375	−1.76120
$86$	0	0
$87$	9.08516	0.974032
$88$	0	0
$89$	−5.69578	−0.603751	−0.301876	−	0.953347i	$-0.597613\pi$
$89$	−5.69578	−0.603751	−0.301876	+	0.953347i	$0.597613\pi$
$90$	0	0
$91$	3.56048	0.373240
$92$	0	0
$93$	3.53465	0.366527
$94$	0	0
$95$	−13.9993	−1.43629
$96$	0	0
$97$	14.9306	1.51597	0.757987	−	0.652270i	$-0.226183\pi$
$97$	14.9306	1.51597	0.757987	+	0.652270i	$0.226183\pi$
$98$	0	0
$99$	−0.732051	−0.0735739

Currently showing only $a_p$; display all $a_n$ Currently showing all $a_n$; display only $a_p$

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4232.2.a.u.1.4	yes	4
4.3	odd	2		8464.2.a.bn.1.2		4
23.22	odd	2		4232.2.a.s.1.3	✓	4
92.91	even	2		8464.2.a.bl.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4232.2.a.s.1.3	✓	4	23.22	odd	2
4232.2.a.u.1.4	yes	4	1.1	even	1	trivial
8464.2.a.bl.1.1		4	92.91	even	2
8464.2.a.bn.1.2		4	4.3	odd	2

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Classical	Maass
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 4232 = 2^{3} \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4232.a (trivial)

\(f(q)\)	\(=\)	\(q+1.41421 q^{3} +2.51764 q^{5} -0.732051 q^{7} -1.00000 q^{9} +0.732051 q^{11} -4.86370 q^{13} +3.56048 q^{15} -6.44949 q^{17} -5.56048 q^{19} -1.03528 q^{21} +1.33850 q^{25} -5.65685 q^{27} +6.42418 q^{29} +2.49938 q^{31} +1.03528 q^{33} -1.84304 q^{35} +2.74202 q^{37} -6.87832 q^{39} +2.46410 q^{41} +2.02922 q^{43} -2.51764 q^{45} -8.71279 q^{47} -6.46410 q^{49} -9.12096 q^{51} +3.08933 q^{53} +1.84304 q^{55} -7.86370 q^{57} -11.4999 q^{59} -2.56753 q^{61} +0.732051 q^{63} -12.2450 q^{65} -13.5347 q^{67} -13.2414 q^{71} -7.26670 q^{73} +1.89293 q^{75} -0.535898 q^{77} +17.1210 q^{79} -5.00000 q^{81} +5.50543 q^{83} -16.2375 q^{85} +9.08516 q^{87} -5.69578 q^{89} +3.56048 q^{91} +3.53465 q^{93} -13.9993 q^{95} +14.9306 q^{97} -0.732051 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{5} + 4 q^{7} - 4 q^{9} - 4 q^{11} - 4 q^{13} - 4 q^{15} - 16 q^{17} - 4 q^{19} + 4 q^{25} - 8 q^{29} - 8 q^{31} + 8 q^{35} - 24 q^{37} - 8 q^{39} - 4 q^{41} - 8 q^{45} - 8 q^{47} - 12 q^{49}+ \cdots + 4 q^{99}+O(q^{100}) \) 4 * q + 8 * q^5 + 4 * q^7 - 4 * q^9 - 4 * q^11 - 4 * q^13 - 4 * q^15 - 16 * q^17 - 4 * q^19 + 4 * q^25 - 8 * q^29 - 8 * q^31 + 8 * q^35 - 24 * q^37 - 8 * q^39 - 4 * q^41 - 8 * q^45 - 8 * q^47 - 12 * q^49 - 8 * q^55 - 16 * q^57 - 8 * q^59 - 4 * q^63 - 16 * q^65 - 32 * q^67 - 16 * q^75 - 16 * q^77 + 32 * q^79 - 20 * q^81 + 8 * q^83 - 44 * q^85 + 24 * q^87 - 16 * q^89 - 4 * q^91 - 8 * q^93 + 16 * q^97 + 4 * q^99

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